| L(s) = 1 | + 4·4-s − 20·7-s + 60·13-s + 12·16-s − 80·28-s + 96·31-s − 60·37-s + 200·49-s + 240·52-s + 284·61-s + 32·64-s − 320·67-s + 320·73-s − 44·79-s − 81·81-s − 1.20e3·91-s − 320·97-s − 20·103-s + 416·109-s − 240·112-s + 384·124-s + 127-s + 131-s + 137-s + 139-s − 240·148-s + 149-s + ⋯ |
| L(s) = 1 | + 4-s − 2.85·7-s + 4.61·13-s + 3/4·16-s − 2.85·28-s + 3.09·31-s − 1.62·37-s + 4.08·49-s + 4.61·52-s + 4.65·61-s + 1/2·64-s − 4.77·67-s + 4.38·73-s − 0.556·79-s − 81-s − 13.1·91-s − 3.29·97-s − 0.194·103-s + 3.81·109-s − 2.14·112-s + 3.09·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 1.62·148-s + 0.00671·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.573216344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.573216344\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 480 T^{2} + p^{4} T^{4} \) |
| good | 5 | $C_2^3$ | \( 1 - 1249 T^{4} + p^{8} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 10 T + 50 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 17807 T^{4} + p^{8} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 15 T + p^{2} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 697 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 178993 T^{4} + p^{8} T^{8} \) |
| 29 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 1680 T^{2} + p^{4} T^{4} )( 1 + 1680 T^{2} + p^{4} T^{4} ) \) |
| 31 | $C_2^2$ | \( ( 1 - 48 T + 1152 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 30 T + 450 T^{2} + 30 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 - 5537953 T^{4} + p^{8} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 1927 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 4290 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 1010 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 2912 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 120 T + p^{2} T^{2} )^{2}( 1 - 22 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 80 T + p^{2} T^{2} )^{4} \) |
| 71 | $C_2^3$ | \( 1 + 21120962 T^{4} + p^{8} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 160 T + 12800 T^{2} - 160 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 22 T + 242 T^{2} + 22 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 1710 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 15392 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 160 T + 12800 T^{2} + 160 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.03692102136504986085300734804, −9.655930444269371025816592530332, −9.571772537409794473480195161627, −9.108931988421568105722706446159, −8.587683349621995591057167829528, −8.524598434370390393591518002996, −8.385026045043970204470622962379, −8.186428782327615660859119976536, −7.40638736500033009195031220195, −7.12777077591844217081655235600, −6.66639127091138834498904303398, −6.64696842506397714282667681152, −6.33118360184808102398740342886, −6.00737550732206080263679791704, −5.97831720030669811736632967809, −5.57805414377180645613688498533, −4.95888642559164610483740493320, −4.20783482611793651290942358465, −3.78525908697981197875014116100, −3.65245340818133119550821327250, −3.24803283848184328429008995745, −2.96884175013973575601260425664, −2.35953668257257693411141766632, −1.35577251202211552047011528125, −0.855589344338345334234829320450,
0.855589344338345334234829320450, 1.35577251202211552047011528125, 2.35953668257257693411141766632, 2.96884175013973575601260425664, 3.24803283848184328429008995745, 3.65245340818133119550821327250, 3.78525908697981197875014116100, 4.20783482611793651290942358465, 4.95888642559164610483740493320, 5.57805414377180645613688498533, 5.97831720030669811736632967809, 6.00737550732206080263679791704, 6.33118360184808102398740342886, 6.64696842506397714282667681152, 6.66639127091138834498904303398, 7.12777077591844217081655235600, 7.40638736500033009195031220195, 8.186428782327615660859119976536, 8.385026045043970204470622962379, 8.524598434370390393591518002996, 8.587683349621995591057167829528, 9.108931988421568105722706446159, 9.571772537409794473480195161627, 9.655930444269371025816592530332, 10.03692102136504986085300734804
